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Stability of a Cartesian Grid Projection Method for Zero Froude Number Shallow Water Flows

Please always quote using this URN: urn:nbn:de:0297-zib-9562
  • In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicola{\"i}des (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered to the computation of incompressible variable density flow problems.

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Metadaten
Author:Stefan Vater, Rupert Klein
Document Type:ZIB-Report
Tag:incompressible flows; inf-sup-condition; mixed finite elements; projection method; shallow water equations; stability
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M12 Stability and convergence of numerical methods
76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Mxx Basic methods in fluid mechanics [See also 65-XX] / 76M12 Finite volume methods
Date of first Publication:2007/05/25
Series (Serial Number):ZIB-Report (07-13)
ZIB-Reportnumber:07-13
Published in:Appeared in: Numerische Mathematik: Volume 113, Issue 1 (2009), Pages 123-161
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